Find the region of convergence for sum from n=1
Find the region of convergence for $\displaystyle\sum_{n=1}^{\infty}\left(1+\frac12+\cdots+\frac1n\right)x^n$.
Explanation of Method
The coefficients are the harmonic numbers $H_n\sim\ln n+\gamma\to\infty$. Since $H_{n+1}/H_n\to1$, the ratio test gives radius $1$; and since $H_n\to\infty$ rather than $0$, the Divergence Test rules out convergence at both endpoints immediately.
- Apply the ratio test, using $H_n\sim\ln n$.
- Test $x=1$: terms $H_n\to\infty$.
- Test $x=-1$: terms oscillate with magnitude $\to\infty$.
Step 1 · Radius of convergence
Since $H_n\sim\ln n+\gamma\to\infty$: $$ \frac{H_{n+1}}{H_n} = 1+\frac{1/(n+1)}{H_n} \longrightarrow 1 $$ (because $H_n\to\infty$ makes the correction term vanish). So $R=1$.
Step 2 · Endpoint $x=1$
$\sum H_n$ has terms $H_n\to\infty\ne0$, so it diverges by the Divergence Test.
Step 3 · Endpoint $x=-1$
The harmonic numbers are strictly increasing ($H_{n+1}=H_n+1/(n+1)>H_n$), so the Leibniz test cannot apply — and more directly, since $|(-1)^nH_n|=H_n\to\infty\ne0$, the terms do not tend to $0$, so the series diverges by the Divergence Test.
Summary Table
| Property | Value |
|---|---|
| Radius of convergence | $R=1$ |
| At $x=1$ | Diverges ($H_n\to\infty$) |
| At $x=-1$ | Diverges (terms don't $\to0$) |
| Region of convergence | $(-1,1)$ |
Additional Notes
- Unlike Problem 232's $\sum(\ln n/n)x^n$, whose coefficients decay to $0$ and admit a nontrivial conditional-convergence endpoint, here the harmonic numbers grow (albeit slowly, like $\ln n$), so neither endpoint has any chance of convergence — a useful contrast between coefficients that grow logarithmically versus decay logarithmically-over-$n$.
- The closed form $\sum H_nx^n=-\ln(1-x)/(1-x)$ for $|x|<1$ makes the divergence at $x=1$ visually obvious: both factors blow up as $x\to1^-$, reinforcing the Divergence-Test conclusion from a different angle.
- This generating function identity is derived by swapping the order of summation in $\sum_n x^n\sum_{k\le n}1/k=\sum_k(1/k)\sum_{n\ge k}x^n$, the same double-sum-reindexing technique used in Problem 226's endpoint evaluations.